6 ms·

I find that the most elegant way to understand exponential-related things is by defining the exponential as the function satisfying f'=f (with suitable normaliz

by sgdpk 6y ago

I find that the most elegant way to understand exponential-related things is by defining the exponential as the function satisfying f'=f (with suitable normalization). In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc. With this definition, you can define the constants e and pi in a natural way. In particular, 2 * pi * i is the period of e^x. It then makes sense to analyse the function e^ix, with period 2*pi, and separate its real and imaginary parts. These are sure to be periodic real functions, and we can name them cos(x) and sin(x). This Quora answer talks about this approach pretty well [1]. [1] https://qr.ae/pNy7qp https://qr.ae/pNy7qp

throwaway77384 6y ago
Here's a fun perspective from someone who knows very little about maths, but has a huge interest in logic. I have a question for you: At what 'level' of maths or understanding do you reckon someone has to be in order to make sense of your introductory paragraph? Let me map out what happens in my head when I read that first paragraph: > "I find that the most elegant way to understand exponential-related things"... Okay, I know what exponential increase means. Something grows by an exponent of something. 2416... > "is by defining the exponential as the function satisfying f'=f (with suitable normalization)"... I don't know what an exponential is. A function, to me, is something I call as part of code, so I am not sure if I might be confusing the programmer and mathematician-definition of 'function'. I don't really know how to interpret f'=f in any way. (F-apostrophe equals f? Why?). I also don't really know how normalization plays into it. For me, normalization is something related to homogenising a series of things. Sometimes it can be related to normalising the volume of a bunch of mp3 files, or in other cases it can be turning a bunch of badly formatted phone numbers into a common format for a database. > "In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc." I have often heard terms like "eigen"-something, but this always causes me trouble, because I end up parsing it in German ("eigen" means "self" in German). So now it is the "self"-function of something? Okay, of the 'derivative'. Is a derivative the mathematician's way of saying "figuring out something you don't know by using a bunch of other things you do know"? I do not know what a linear operator is at all. I also do not know what ODEs are. You see, I love logic and science and have been a programmer for so long, yet when I hear something like the statement above, I almost become despondent over the fact that seemingly I know nothing of the underpinnings or logical foundations of the things I spend my life with (aka computers and programming being born from maths...). Anyway, just a random drive-by thought.
- gxs 6y ago
  Tangentially related to your point - math is an extremely dense subject. Small little statements with just a few symbols can be LOADED with information - things like complex, meaty definitions for even the most benign terms (e.g., a function), and stuff that can be deduced from it from a million different theorems. This is part of the reason why it's so easy to fall behind in math. Once you are missing a vital part of information, it can be hard to identify where that gap is and the train just keeps moving and moving. This was my experience with math, and something that I noticed when I tutored people in some low level math as well.
  - hansvm 6y ago
    I've thought about this on occasion. Anecdotally (and subject to selection and confirmation biases) I've seen students who struggled excessively in a given course, dropped, and taken the previous course to excel not just in the subject they originally dropped but also in subsequent classes. I've wondered if my alma mater might be better off recommending students to start a level or two below where they currently are to fill in some of those vital gaps.
- sgdpk 6y ago
  Sure, there is certainly a lot of jargon here which I could have explained better, but in the end one has to be familiar with some calculus and complex numbers to make sense of this topic. Depending on where you're from, this could be either late high-school to second/third year of university. So by exponential I mean the exponential function, in the mathematician's sense: something which takes numbers as inputs and produces other numbers as outputs in a unique way. Already a complication: these can be complex numbers. Then, the derivative of a function is its "instantaneous rate of change". How much does the function increase for small (infinitesimal, really) changes in input? Since this rate may be different for every point, we are defining a new function f': for every input of f, f' gives you the local rate of change. So there you go, the derivative takes a function f and produces a function f'. We call this an operator. Because the derivative has very nice properties, we call this a "linear" operator. Finally, the "eigen"-things. These are really fancy words for the simple concept of a fixed-point: when you apply an operation to a certain thing X and just get that X back. So, the function x^2 has two fixed-points: 0 and 1. These would be "eigenpoints", but we don't call them that because we reserve the term "eigen" for linear operations, such as the derivative operator or matrix-multiplication. The point of my answer is that the exponential function is elegantly understood as the eigenfunction of the derivative. In other words: which function is its own derivative? Which one tells you how it changes just by looking at it?
gxs 6y ago
That was an EXCELLENT read, thank you. I was a math major in college, and somethings you learn how to work out mechanically and understand intellectually. I dont do anything remotely close to academic, undergraduate mathematics, but it's interesting how now that I'm older i seem to understand things better, to internalize them beyond something mechanical or simply accepting them because they make logical sense. Wish I'd had that type of understanding in school. This explanation is very intuitive - and the author makes a good point. There might be better ways to introduce pi other than the ratio of the circumference of a circle to its diameter.
- jonsen 6y ago
  > There might be better ways to introduce pi other than the ration of the circumference of a circle to it's radius. Sure, the ratio of the circumference of a circle to it's diameter ;-)
  - gxs 6y ago
    Ha! I thought I corrected it before anyone saw :)
- qsort 6y ago
  > There might be better ways to introduce pi To be fair that's the elementary school way of defining pi. I'm somewhat rusty myself, but I'm pretty sure even undergraduate real analysis courses define pi in terms of the complex exponential. "Ratio of circumference to diameter" is insanely hard to make rigorous.
  - Tainnor 6y ago
    My undergraduate real analysis course didn't, probably because here, I don't think complex numbers are necessarily part of the high school curriculum. Instead, we defined sin and cos via power series, and then IIRC, pi as the first positive zero of the sin function or something similar.
- sgdpk 6y ago
  Indeed! I highly recommend following the author.

by sgdpk 6y ago

throwaway77384 6y ago

Here's a fun perspective from someone who knows very little about maths, but has a huge interest in logic. I have a question for you: At what 'level' of maths or understanding do you reckon someone has to be in order to make sense of your introductory paragraph? Let me map out what happens in my head when I read that first paragraph: > "I find that the most elegant way to understand exponential-related things"... Okay, I know what exponential increase means. Something grows by an exponent of something. 2416... > "is by defining the exponential as the function satisfying f'=f (with suitable normalization)"... I don't know what an exponential is. A function, to me, is something I call as part of code, so I am not sure if I might be confusing the programmer and mathematician-definition of 'function'. I don't really know how to interpret f'=f in any way. (F-apostrophe equals f? Why?). I also don't really know how normalization plays into it. For me, normalization is something related to homogenising a series of things. Sometimes it can be related to normalising the volume of a bunch of mp3 files, or in other cases it can be turning a bunch of badly formatted phone numbers into a common format for a database. > "In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc." I have often heard terms like "eigen"-something, but this always causes me trouble, because I end up parsing it in German ("eigen" means "self" in German). So now it is the "self"-function of something? Okay, of the 'derivative'. Is a derivative the mathematician's way of saying "figuring out something you don't know by using a bunch of other things you do know"? I do not know what a linear operator is at all. I also do not know what ODEs are. You see, I love logic and science and have been a programmer for so long, yet when I hear something like the statement above, I almost become despondent over the fact that seemingly I know nothing of the underpinnings or logical foundations of the things I spend my life with (aka computers and programming being born from maths...). Anyway, just a random drive-by thought.

gxs 6y ago
Tangentially related to your point - math is an extremely dense subject. Small little statements with just a few symbols can be LOADED with information - things like complex, meaty definitions for even the most benign terms (e.g., a function), and stuff that can be deduced from it from a million different theorems. This is part of the reason why it's so easy to fall behind in math. Once you are missing a vital part of information, it can be hard to identify where that gap is and the train just keeps moving and moving. This was my experience with math, and something that I noticed when I tutored people in some low level math as well.
- hansvm 6y ago
  I've thought about this on occasion. Anecdotally (and subject to selection and confirmation biases) I've seen students who struggled excessively in a given course, dropped, and taken the previous course to excel not just in the subject they originally dropped but also in subsequent classes. I've wondered if my alma mater might be better off recommending students to start a level or two below where they currently are to fill in some of those vital gaps.
sgdpk 6y ago
Sure, there is certainly a lot of jargon here which I could have explained better, but in the end one has to be familiar with some calculus and complex numbers to make sense of this topic. Depending on where you're from, this could be either late high-school to second/third year of university. So by exponential I mean the exponential function, in the mathematician's sense: something which takes numbers as inputs and produces other numbers as outputs in a unique way. Already a complication: these can be complex numbers. Then, the derivative of a function is its "instantaneous rate of change". How much does the function increase for small (infinitesimal, really) changes in input? Since this rate may be different for every point, we are defining a new function f': for every input of f, f' gives you the local rate of change. So there you go, the derivative takes a function f and produces a function f'. We call this an operator. Because the derivative has very nice properties, we call this a "linear" operator. Finally, the "eigen"-things. These are really fancy words for the simple concept of a fixed-point: when you apply an operation to a certain thing X and just get that X back. So, the function x^2 has two fixed-points: 0 and 1. These would be "eigenpoints", but we don't call them that because we reserve the term "eigen" for linear operations, such as the derivative operator or matrix-multiplication. The point of my answer is that the exponential function is elegantly understood as the eigenfunction of the derivative. In other words: which function is its own derivative? Which one tells you how it changes just by looking at it?

gxs 6y ago

That was an EXCELLENT read, thank you. I was a math major in college, and somethings you learn how to work out mechanically and understand intellectually. I dont do anything remotely close to academic, undergraduate mathematics, but it's interesting how now that I'm older i seem to understand things better, to internalize them beyond something mechanical or simply accepting them because they make logical sense. Wish I'd had that type of understanding in school. This explanation is very intuitive - and the author makes a good point. There might be better ways to introduce pi other than the ratio of the circumference of a circle to its diameter.

jonsen 6y ago
> There might be better ways to introduce pi other than the ration of the circumference of a circle to it's radius. Sure, the ratio of the circumference of a circle to it's diameter ;-)
- gxs 6y ago
  Ha! I thought I corrected it before anyone saw :)
qsort 6y ago
> There might be better ways to introduce pi To be fair that's the elementary school way of defining pi. I'm somewhat rusty myself, but I'm pretty sure even undergraduate real analysis courses define pi in terms of the complex exponential. "Ratio of circumference to diameter" is insanely hard to make rigorous.
- Tainnor 6y ago
  My undergraduate real analysis course didn't, probably because here, I don't think complex numbers are necessarily part of the high school curriculum. Instead, we defined sin and cos via power series, and then IIRC, pi as the first positive zero of the sin function or something similar.
sgdpk 6y ago
Indeed! I highly recommend following the author.